Sum of Series Calculator
Solve sum series problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Calculator
Adjust values & calculatePartial Sums
Formula
For arithmetic series, S = n/2 times (first + last term). For geometric series, S = a(1 - r^n)/(1 - r) where r is the common ratio. For convergent infinite geometric series (|r| < 1), S = a/(1 - r).
Last reviewed: December 2025
Worked Examples
Example 1: Arithmetic Series Sum
Example 2: Infinite Geometric Series
Background & Theory
The Sum of Series Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Sum of Series Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
Arithmetic: S = n/2(a+l) | Geometric: S = a(1-r^n)/(1-r)
For arithmetic series, S = n/2 times (first + last term). For geometric series, S = a(1 - r^n)/(1 - r) where r is the common ratio. For convergent infinite geometric series (|r| < 1), S = a/(1 - r).
Worked Examples
Example 1: Arithmetic Series Sum
Problem: Find the sum of the first 20 terms of the arithmetic series 5 + 8 + 11 + 14 + ...
Solution: a = 5, d = 3, n = 20\nLast term l = a + (n-1)d = 5 + 19(3) = 62\nS = n/2 x (a + l) = 20/2 x (5 + 62) = 10 x 67 = 670
Result: Sum = 670 | Last term = 62 | Average term = 33.5
Example 2: Infinite Geometric Series
Problem: Find the sum of the infinite geometric series 12 + 6 + 3 + 1.5 + ...
Solution: a = 12, r = 0.5 (|r| < 1, so the series converges)\nFinite sum of 10 terms: S10 = 12(1 - 0.5^10)/(1 - 0.5) = 12(0.999023)/0.5 = 23.977\nInfinite sum: S = a/(1-r) = 12/(1-0.5) = 12/0.5 = 24
Result: Finite sum (10 terms) = 23.977 | Infinite sum = 24
Frequently Asked Questions
What is a mathematical series and how is it different from a sequence?
A sequence is an ordered list of numbers following a pattern, while a series is the sum of the terms in a sequence. For example, 1, 3, 5, 7, 9 is a sequence; 1 + 3 + 5 + 7 + 9 = 25 is the corresponding series. Sequences describe individual terms, while series describe cumulative totals. A finite series sums a specific number of terms, while an infinite series sums infinitely many terms and may or may not converge to a finite value. The distinction matters because a sequence can have well-defined terms without its series converging. Understanding both concepts is fundamental to calculus, analysis, and applied mathematics.
What is an arithmetic series and what is its sum formula?
An arithmetic series is the sum of terms in an arithmetic sequence, where each term differs from the previous by a constant called the common difference (d). The sum of n terms is S = n/2 times (first term + last term), equivalently S = n/2 times (2a + (n-1)d). This formula works because pairing the first term with the last, the second with the second-to-last, and so on produces equal pairs. The famous story of young Gauss summing 1 to 100 exemplifies this: 50 pairs each summing to 101 gives 5050. Arithmetic series appear in financial calculations like total payments on a uniformly increasing schedule and in physics for uniformly accelerated motion.
What is a geometric series and when does it converge?
A geometric series is the sum of terms in a geometric sequence, where each term is multiplied by a constant ratio (r) to get the next term. The finite sum of n terms is S = a(1 - r^n) / (1 - r), where a is the first term. An infinite geometric series converges (has a finite sum) only when the absolute value of r is less than 1, in which case the sum equals a / (1 - r). For example, 1 + 1/2 + 1/4 + 1/8 + ... = 1 / (1 - 0.5) = 2. When the absolute value of r is 1 or greater, the infinite series diverges. Geometric series are fundamental in finance (present value of annuities), physics (bouncing ball total distance), and signal processing.
What is a harmonic series and why is it important?
The harmonic series is the sum 1 + 1/2 + 1/3 + 1/4 + 1/5 + ..., where each term is the reciprocal of a natural number. Despite its terms approaching zero, the harmonic series diverges (grows without bound), though extremely slowly. After summing one million terms, the total is only about 14.39. This counterintuitive result is important because it shows that terms going to zero is necessary but not sufficient for series convergence. The harmonic series appears in probability (coupon collector problem), music theory (overtone frequencies), and computer science (analysis of algorithms). The partial sums of the harmonic series approximate ln(n) + 0.5772 (Euler-Mascheroni constant).
How do you determine if an infinite series converges or diverges?
Several tests help determine convergence. The divergence test says if terms do not approach zero, the series diverges. The ratio test compares consecutive terms: if the limit of |a(n+1)/a(n)| is less than 1, the series converges absolutely. The root test checks if the nth root of |a(n)| has a limit less than 1. The comparison test compares to a known series. The integral test connects series convergence to improper integrals. The alternating series test applies when signs alternate and terms decrease toward zero. For geometric series, convergence requires |r| < 1. No single test works for all series, and choosing the right test is a key skill in calculus courses.
What are power series and how are they used?
A power series is an infinite series of the form sum of a_n times x^n, where x is a variable and a_n are coefficients. Every power series has a radius of convergence R, within which it converges and outside which it diverges. Taylor and Maclaurin series are power series that represent functions. For example, e^x = 1 + x + x^2/2! + x^3/3! + ..., sin(x) = x - x^3/3! + x^5/5! - ..., and 1/(1-x) = 1 + x + x^2 + x^3 + ... for |x| < 1. Power series are essential in physics for approximating solutions to differential equations, in numerical computing for evaluating transcendental functions, and in engineering for analyzing systems near equilibrium.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy